This week

The description of subspaces invariant under the Volterra operator goes back to a problem of Gelfand from 1938. Invariant subspaces for differentiation on $C^{\infty}$ were studied much later by Aleman and Korenblum and continued by Aleman, Baranov and Belov. Both problems contain a wealth of interesting ideas and have several interesting connections to exponential systems, among other things. I intend to give a review of some of these results and then continue with a more abstract setting consisting of an unbounded operator D with a compact quasi-nilpotent right inverse V. It turns out that under certain general conditions one can prove similar results for a large class of examples (for D) containing Schrödinger operators, Dirac operators and other Canonical systems of differential equations. This is a report about recent joint work with Alexandru Aleman.